Elements of Algebra

CHAP. VI. OF ALGEBRA. 349 purpose. But if from those values we pass to the succeeding ones, we constantly obtain integer numbers; which, indeed, we should have obtained much more easily by means of the numbers $p$ and $q$, that were introduced at the beginning. In fact, if we take $p$ and $q$, so that $p^2 = aq^2 + 1$, we shall have $aq^2 - p^2 = -1$, and the fractions will disappear. For then $x = -2gpq + f(aq^2 + p^2) + bq^2$, and $y = -g(aq^2 + p^2) + 2afpq + bpq$; but as in the known case, $af^2 + bf + c = g^2$, we find only the second power of $g$, it is of no consequence what sign we give that letter; if, therefore, we write $-g$, instead of $+g$, we shall have the formulæ $x = 2gpq + f(aq^2 + p^2) + bq^2$, and $y = g(aq^2 + p^2) + 2afpq + bpq$, and we shall thus be certain, at the same time, that $ax^2 + bx + c = y^2$. Let it be required, as an example, to find the hexagonal numbers that are also squares. We must have $2x^2 - x = y^2$, or $a=2$, $b=-1$, and $c=0$, and the known case will evidently be $x=f=1$, and $y=g=1$. Farther, in order that we may have $p^2 = 2q^2 + 1$, we must have $q=2$, and $p=3$; so that we shall have $x = 12g + 17f - 4$, and $y = 17g + 24f - 6$; whence result the following values: $x=f=1$ | $25$ | $841$ $y=g=1$ | $35$ | $1189, \ldots$ 94. Let us also consider our first formula, in which the second term was wanting, and examine the cases which make the formula $ax^2 + b$ a square in integer numbers. Let $ax^2 + b = y^2$, and it will be required to fulfil two conditions: 1. We must know a case in which this equation exists; and we shall suppose that case to be expressed by the equation $af^2 + b = g^2$. 2. We must know such values of $m$ and $n$, that $m^2 = an^2 + 1$; the method of finding which will be taught in the next chapter. From that results a new case, namely, $x = ng + mf$, and $y = mg + anf$; this, also, will lead us to other similar cases, which we shall represent in the following manner: $x=f$ | $A$ | $B$ | $C$ | $D$ | $E$ $y=g$ | $P$ | $Q$ | $R$ | $S$ | $T, \ldots$ In which, $A=ng+mf$ | $B=nP+mA$ | $C=nQ+mB$ | $D=nR+mC$ | $E=nS+mD$ $P=mg+anf$ | $Q=mP+anA$ | $R=mQ+anB$ | $S=mR+anC$ | $T=mS+anD, \ldots$